Predicting Survival Probability of NASA Aircraft Engines

Using Cox Proportional Hazards Model

Jayme Reed & Brad Paton (Advisor: Dr. Cohen)

April 15, 2025

Cox Proportional Hazards (CPH) Model

What is it?

  • A statistical regression method specializing in modeling time-to-event predictions with survival data (Abeysekera and Sooriyarachchi 2009)

    • Survival data has a value for time and an indicator column for an event

  • Is a method that can deal with censored data

    • Censored data is when the information about an individual in a study is only known for a certain period of time (Klein and Moeschberger 2005)

  • Primarily used in the health field but has applications in predicting bank failure, the survival probability of machines, and insurance likelihood payouts

Limitations

Mathematical Formulas

  • Survival Function: \(S(x) = \int_{x}^{\infty} f(x)dx\)

  • Hazard Function: \(h(x) = \lim\limits_{\Delta x \rightarrow 0} \frac{P[x \leq X < x + \Delta x|X \geq x]}{\Delta x}\)

  • CPH Model Hazard Function: \(h(t|\mathbf Z) = h_0(t)\text{exp}(\sum\limits_{k=1}^{p} \beta_kZ_k)\)

  • Proportional Hazards Ratio: \(\frac{h(t|\mathbf Z)}{h(t|\mathbf Z*)} = \text{exp}[\sum\limits_{k=1}^{p} \beta_k(Z_k - Z_k^*)]\)

  • Cumulative Hazard Function: \(H(x) = \int_0^x h(u) du\)

  • Concordance Index: \(C = \frac{c + \frac{t_x}{2}}{c + d + t_x}\)

  • Survival Probability: \(S(t) = e^{-H(t)}\), where \(H(t)\) is the above cumulative hazard function

Assumptions

There are four assumptions for CPH:

  • Independence assumption

    • Assumes that the survival times of observed subjects are independent of each other (Nahhas 2025)

  • Non-informative Censoring Assumption

    • Assumes that censoring is non-informative (Nahhas 2025)

  • Linearity Assumption

    • Assumes the relationship between covariates and the outcome is a linear relationship (Nahhas 2025)

  • Proportional Hazards Assumption

    • Assumes the ratio of hazards rates for any two subjects are constant at all times (Bustan 2018)

Linearity Assumption

  • Tested using the Martingale residuals for each covariate using the equation Martingale Residuals = Observed Events - Expected Results (Nahhas 2025)

  • If the plots are linear and appear to have a slope of zero, the assumption is not violated (Amini 2015)

Proportional Hazards Assumption

  • The test is verified using the Schoenfeld partial residuals which are the difference between the value of the covariate and the expected value of the covariate at the time of failure (Klein and Moeschberger 2005)

variable chisq df p
meal.cal 4.0478025 1 0.0442289
age 0.4838469 1 0.4866850
GLOBAL 4.1100198 2 0.1280916

Kaplan-Meier Survival Curve

  • Provides visualization for the Kaplan-Meier estimator

  • Is considered to be well defined up until the largest observed study time \(t_{max}\)

  • Demonstrates the time where the event being modeled is expected to occur

Forest Plot

  • Provides visualization of the effects of each covariate on the hazard ratio

  • A positive effect indicates a positive correlation with the hazard ratio

Evaluation

Prediction

Data Structure, Exploration, and Visualization

Creating CPH Model

Checking Assumptions

Adjusting the Model

Model Results

Conclusion

References

Abeysekera, W. W. M., and M. R. Sooriyarachchi. 2009. “Use of Schoenfeld’s Global Test to Test the Proportional Hazards Assumption in the Ox Proportional Hazards Model: An Application to a Clinical Study.” https://www.researchgate.net/publication/238483310_Use_of_Schoenfeld's_global_test_to_test_the_proportional_hazards_assumption_in_the_Cox_proportional_hazards_model_An_application_to_a_clinical_study.
Amini, Zaki. 2015. “Log-Linearity for Cox’s Regression Model.” https://www.duo.uio.no/bitstream/handle/10852/45377/thesis_zaki.pdf?sequence=15&isAllowed=y.
Asghar, Naseem, Khalil Umair, and Iftikhar Uddin. 2024. “Mixture and Non-Mixture Cure Models for the Survival Analysis of SARS-CoV-2 Patients in Khyber Pakhtunkhwa, Pakistan.” Pakistan Journal of Medical Sciences 40 (8): 1841–46.
Bustan, M. Nadjib. 2018. “Cox Proportional Hazard Survival Analysis to Inpatient Breast Cancer Cases.”
Jiang, Nan, Wu Yongfa, and Chengjia Li. 2024. “Limitations of Using COX Proportional Hazards Model in Cardiovascular Research.” Cardiovascular Diabetology, no. 1: 1–2.
Klein, John P., and Melvin L. Moeschberger. 2005. Survival Analysis: Techniques for Censored and Truncated Data. 2nd ed. Springer.
Nahhas, Ramzi W. 2025. Introduction to Regression Methods for Public Health Using r. 1st ed. Chapman & Hall.
Wang, Weiwei, Xiaotian Chang, and Feifei Lin. 2025. “Adding Salt to Foods and Risk of Incident Depression and Anxiety.” BMC Medicine, no. 1: 1–10.
Zhang, Yue, Yangyang Cheng, and Rodrigo M Carrillo-Larco. 2025. “Postpartum Depression in Relation to Chronic Diseases and Multimorbidity in Women’s Mid-Late Life: A Prospective Cohort Study of UK Biobank.” BMC Medicine 23 (1): 1–13.